PHASE FIELD MODELS FOR THIN ELASTIC ...cvgmt.sns.it/media/doc/paper/2737/thin_elastic_no_penal.pdfPHASE FIELD MODELS FOR THIN ELASTIC STRUCTURES WITH TOPOLOGICAL CONSTRAINT PATRICK

PHASE FIELD MODELS FOR THIN ELASTIC STRUCTURES WITH

TOPOLOGICAL CONSTRAINT

PATRICK W. DONDL, ANTOINE LEMENANT, AND STEPHAN WOJTOWYTSCH

Abstract. This article is concerned with the problem of minimising the Willmore energy in the

class of connected surfaces with prescribed area which are confined to a container. We propose

a phase field approximation based on De Giorgi’s diffuse Willmore functional to this variationalproblem. Our main contribution is a penalisation term which ensures connectedness in the sharp

interface limit.

For sequences of phase fields with bounded diffuse Willmore energy and bounded area term,we prove uniform convergence in two ambient space dimensions and a certain weak mode of

convergence on curves in three dimensions. This enables us to show Γ-convergence to a sharpinterface problem that only allows for connected structures. The topological contribution is based

on a geodesic distance chosen to be small between two points that lie on the same connected

component of the transition layer of the phase field.We furthermore present numerical evidence of the effectiveness of our model. The implemen-

tation relies on a coupling of Dijkstra’s algorithm in order to compute the topological penalty

to a finite element approach for the Willmore term.

Contents

1. Introduction 22. Phase Fields 52.1. General Background 52.2. Equipartition of Energy 62.3. Connectedness 62.4. Main Results 73. Proofs 83.1. Proof of Γ-Convergence 93.2. Auxiliary Results I 103.3. Auxiliary Results II 173.4. Proof of the Main Results 244. Computer Implementation 275. Conclusions 29Conflict of Interest 31References 31

Date: July 10, 2015.2010 Mathematics Subject Classification. 49Q10; 74G65; 65M60.Key words and phrases. Willmore energy, phase field approximation, topological constraint, Gamma-

convergence.

1

2 PATRICK W. DONDL, ANTOINE LEMENANT, AND STEPHAN WOJTOWYTSCH

1. Introduction

In this article, we consider a diffuse interface approximation of a variational problem arisingin the study of thin elastic structures. Our particular question is motivated by the problem ofpredicting the shape of certain biological objects, such as mitochondria, which consist of an elas-tic lipid bilayer and are confined by an additional outer mitochondrial membrane of significantlysmaller surface area. When modelling such structures, in addition to the confinement and surfacearea constraints, a challenging topological side condition naturally arises as the biological mem-branes are typically connected and do not self-intersect. Incorporating this property into a diffuseinterface model will be the main concern of this article.

In order to describe the the locally optimal shape of thin elastic structures, we will rely onsuitable bending energies. A well-known example of such a variational characterisation of biomem-branes is given by the Helfrich functional [Hel73, Can70]

(1.1) EH(Σ) = χ0

∫Σ

(H −H0)2 dH2 + χ1

∫Σ

K dH2

where Σ denotes the two-dimensional membrane surface in R3 and H and K denote its mean andGaussian curvatures. The parameters χ0, χ1 and H0 are the bending moduli and the spontaneouscurvature of the membrane. Both integrals are performed with respect to the two-dimensionalHausdorff measure H2. The special case when H0, χ1 = 0 is known as Willmore’s energy

W(Σ) =

∫Σ

H2 dH2.

Extrema (in particular, local minimisers) of the Willmore functional are therefore of interest inmodels for biological membranes, but they also arise naturally in pure differential geometry as thestereographic projections of compact minimal surfaces in S3. An introduction from that point ofview is given in [PS87], along with a number of examples. From the point of view of the calculusof variations, a natural approach to energies as the ones above is via varifolds [All72].

In the class of closed surfaces, the second term in the Helfrich functional is of topological nature.If the minimisation problem is considered only among surfaces of prescribed topological type itcan be neglected due to the Gauss-Bonnet theorem. The spontaneous curvature is realisticallyexpected to be non-zero and can have tremendous influence. It should be noted that the fullHelfrich energy depends also on the orientation of a surfaces for H0 6= 0 and not only on itsinduced (unoriented) varifold. Große-Brauckmann [GB93] gives an example of surfaces Mk ofconstant mean curvature H ≡ 1 converging to a doubly covered plane. This demonstrates that,unlike the Willmore energy, the Helfrich energy need not be lower semi-continuous under varifoldconvergence for some parameters.

Even for Willmore’s energy rigorous results are hard to obtain. The existence of smooth min-imising Tori was proved by Simon [Sim93], and later generalised to surfaces of arbitrary genus in[BK03]. The long-standing Willmore conjecture thatW(T ) ≥ 4π2 for all Tori embedded in R3 wasrecently established in [MN14], and the large limit genus of the minimal Willmore energy for closedorientable surfaces in R3 has been investigated in [KLS10]. The existence of smooth minimisingsurfaces under isoperimetric constraints has been established in [Sch12]. A good account of theWillmore functional in this context can be found in [KS12].

The case of surfaces constrained to the unit ball was studied in [MR14] and a scaling law forthe Willmore energy was found in the regimes of surface area just exceeding 4π and the large arealimit. While the above papers adopt an external approach in the language of varifold geometry, aparametrised approach has been developed in [Riv14] and related papers.

PHASE FIELD MODELS WITH TOPOLOGICAL CONSTRAINT 3

Recently, existence of minimisers for certain Helfrich-type energies among axially symmetricsurfaces under an isoperimetric constraint was proved by Choksi [CV13]. Other avenues of researchconsider Willmore surfaces in more general ambient spaces.

Results for the gradient flow of the Willmore functional are still few. Short time existencefor sufficiently smooth initial data and long time existence for small initial energy have beendemonstrated in [KS01, KS12] and it has been shown that Willmore flow can drive smooth initialsurfaces to self-intersections in finite time in [MS03]. A level set approach to Willmore flow isdiscussed in [DR04].

Studies of numerical implementations of Willmore flow are, for example, due to Garcke, Dziuk,Elliott et al. in [BGN08, Dzi08, DE07]. Particularly interesting here is also an implementation ofa two-step time-discretisation algorithm due to Rumpf and Balzani [BR12].

Phase-field approximations of the functional in (1.1), on the other hand, often provide a moreconvenient approach to gradient flows or minimisation of the Willmore or Helfrich functionals. Suchphase-field models are generally based on a competition between a multi-well functional penalisingdeviation of the phase field function from the minima (usually ±1) and a gradient-penalising termpreventing an overly sharp transition. The prototype is the Modica-Mortola [MM77, Mod87]functional

Sε(u) =1

c0

∫Ω

ε

2|∇u|2 +

1

εW (u) dx,

which is also known as the Cahn-Hilliard energy. Here, W is a function with minimizers at ±1,usually taken to be W (s) = 1

4 (s2−1)2. The normalising constant c0 is taken such that Sε convergesin the sense of Γ-convergence to the perimeter functional of the jump-set of a function that is ±1almost everywhere in the domain Ω. The reason is that a transition in u from +1 to −1 alwaysrequires a certain minimal energy (per unit area of the transition), but this minimal energy can beattained in the limit by an optimal transition (of tanh-shape orthogonal to the interface, in thiscase). In particular if a coupling of the surface to a bulk term is desired, phase field models canprovide an excellent alternative to a parametric discretisation [Du10].

The idea for applying this approach to Willmore’s energy goes back to De Giorgi [DG91]. Fora slight modification of De Giorgi’s functional, reading

EdGε (u) =

1

c0

∫Ω

1

ε

(ε∆u− 1

εW ′(u)

)2

dx + λSε(u) =:Wε(u) + λSε(u)

Γ-convergence to the sum of Willmore’s energy and the λ-fold perimeter functional was finallyproved by Roger and Schatzle [RS06] in n = 2, 3 dimensions, providing the lower energy bound,after Bellettini and Paolini [BP93] had provided the recovery sequence over a decade earlier. Firstanalytic evidence had previously been presented in the form of asymptotic expansions in [DLRW05].This and other phase field approximations of Willmore’s energy and their L2-gradient flows arereviewed in [BMO13]. A convergence result for a diffuse approximation of certain more generalHelfrich-type functionals has also been derived by Belletini and Mugnai [BM10].

Regarding implementation of phase-field models for Willmore’s and Helfrich’s energy, we referto the work by Misbah et al. in [BKM05] and Du et al. [DLRW05, DLW05, DLW06, DW07,DLRW07, DLRW09, Du10, WD07]. In [FRW13], the two-step algorithm for surface evolution hasbeen extended to phase fields. A numerical implementation of the Helfrich functional can be found,for example, in the work by Campelo and Hernandez-Machado [CHM06].

Starting with the above given phase-field approximation of Willmore’s energy, the confinementand the surface area condition are both easily incorporated. The confinement is simply given bythe domain of the phase field function with suitable boundary condition; the surface area of thetransition layer can for example be fixed by an energy term which penalises the deviation of Sεfrom a given target value. This leaves the topological side condition as our main challenge.


An often cited advantage of phase fields is that they are capable of changing their topology;in that sense our endeavour is non-standard. It should be noted that our phase fields may stillchange their topology (at least in three dimensions), only connectedness is enforced. Additionally,non-interpenetration and confinement constraints are equally difficult to enforce in sharp interfaceevolutions. So for the moment we neglect the topological term in the Helfrich functional and focuson the more imminent topological property of connectedness. Still, we remark that our results canbe extended to the diffuse Helfrich functionals from [BM10].

Approaches of regularising limit interfaces have been developed by Bellettini in [Bel97] andinvestigated analytically and numerically in [ERR14]. The approaches work by introducing non-linear terms of the phase-field in order to control the Willmore energies of the level sets individuallyand exclude transversal crossings (which phase fields for De Giorgi’s functional can develop). Theseregularisations may prevent loss of connectedness along a gradient flow in practice, but do not leadto a variational statement via Γ-convergence.

Previous work in [DMR11] provides a first attempt at an implementation of a topological con-straint in a phase-field model for elastic strings modelled by the one-dimensional version of theWillmore energy, namely Euler’s elastica. This approach was complemented by a method putforth in [DMR14], which relies on a second phase field subject to an auxiliary minimisation prob-lem used to identify connected components of the transition layer. For this model, a Γ-convergenceresult was obtained, showing that limits of bounded-energy sequences must describe a connectedstructure. Unfortunately, the complicated nested minimisation problem makes it unsuitable forcomputation.

In this article, we propose a different topological penalty term in the energy functional, inspiredby [BLS15, LS14], which uses a carefully constructed distance function to detect connected com-ponents. The basic idea is that this geodesic distance between two points in the domain shouldbe small if and only if the points can be connected by a curve lying in the phase field transitionregion. Connected components of the transition layer will now have a finite distance, and thus theoccurrence of more than one such component can be sensed by the functional.

Along our proof, we obtain a few useful technical results about the convergence of phase fieldapproximations to the limit problem which appear to be new in this context. Namely, for sequencesalong which EdG

ε remains bounded, we prove convergence of the transition layers to the limit curvein the Hausdorff distance and uniform convergence of the phase field to ±1 away from the limitcurve in two dimensions. This resembles a similar result for minimisers of the Modica-Mortolafunctional among functions with prescribed integral from [CC95] and more generally stationarystates from [HT00].

In three dimensions, we show that neither result is true, but prove weak analogues. We alsoshow that phase fields are uniformly L∞-bounded in terms of their Willmore energy, also in threedimensions.

The paper is organised as follows. In section 2 we give a brief introduction to phase fields forWillmore’s problem in general (sections 2.1 and 2.2) and to our approach to connectedness for thetransition layers (section 2.3). Our main results are subsequently listed in section 2.4.

Section 3 is entirely dedicated to the proofs of our main results. We directly proceed to showΓ-convergence of our functionals (section 3.1). In sections 3.2 and 3.3, we produce all the auxiliaryestimates that will be needed in the later proofs. These are then used to show convergence of thephase-fields, Hausdorff-convergence of the transition layers, and connectedness of the support ofthe limit measure (section 3.4).

In section 4 we briefly show numerical evidence of the effectiveness of our approach. To thatend, we compare it to diffuse Willmore flow without a topological term and to the penalisationproposed in [DMR11]. We have been unable to implement the functional developed in [DMR14]


in practice, so a comparison with that could not be drawn. In section 5 we discuss a few easyextensions of our main results and related open problems.

The numerical implementation of our functional will be discussed further in a forth-comingarticle [DW15]. We use Dijkstra’s algorithm to compute the geodesic distance function used inthe topological term of our energy functional. The weight in the geodesic distance is chosen to beexactly zero along connected components of the transition layer, which implies that it only needsto be computed once per detected connected component. This makes the functional extremelyefficient from an implementation point of view.

2. Phase Fields

2.1. General Background. A phase-field approach is a method which lifts problems of (n− 1)-dimensional manifolds which are the boundaries of sets to problems of scalar fields on n-dimensionalspace. Namely, instead of looking at ∂E, we study smooth approximations of χE − χEc . FixΩ b Rn, open. Then our model space is

X = u ∈W 2,2loc (Rn) | u ≡ −1 outside Ω.

If ∂Ω ∈ C2, this can be identified with Xa = u ∈ W 2,2(Ω) | u = ∂νu = 0 on ∂Ω, otherwise thefirst formulation is technically easier to work with. The boundary conditions are chosen to modelsets which are totally contained in Ω and whose boundaries may only touch ∂Ω tangentially. Then(as well as in more general settings) it is known [Mod87] that the functionals

Sε : L1(Ω)→ R, Sε(u) =

1c0

∫Ωε2 |∇u|

2 + 1ε W (u) dx u ∈ X

+∞ else,

with W (u) = (u2 − 1)2/4 and c0 =∫ 1

−1

√2W (u) du = 2

√2 /3, Γ-converge to

S0(u) =1

2|Du|(Ω)

with respect to strong L1-convergence for functions u ∈ BV (Rn, −1, 1) (and +∞ else). Thelimit S0(u) agrees with the perimeter functional Per(u = 1). The double-well potential Wforces sequences of bounded energy Sε(uε) to converge to a function of the above form as ε →0, and the functional has originally been studied in the context of minimal surfaces, see e.g.[Mod87, CC95, CC06]. We will view them in a slightly different light. If uε is a finite energysequence, we denote the Radon measure given by the approximations of the area functional as

µε(U) =1

c0

∫U

ε

2|∇uε|2 +

1

εW (uε) dx.

Taking the first variation of this measure, and integrating by parts, we get

δµε(φ) =1

c0

∫Ω

(−ε∆uε +

1

εW ′(uε)

)φdx.

If we take the integrand

vε := −ε∆uε +1

εW ′(uε),

we obtain a diffuse analogue of the mean curvature as the variation of surface area. A modifiedversion of a conjecture by De Giorgi then proposes that squaring vε, integrating over Ω and dividingby ε should be a Γ-convergent approximation of W(∂E) for C2-regular sets E, again with respectto strong L1-convergence. Dividing by ε is needed to convert the volume integral into a diffusesurface integral over the transition layer with width of order ε. First analytic evidence for thiswas provided via formal asymptotic expansions in [DLRW05] and the conjecture was proved for


n = 2, 3 in [RS06]. From now on, we will restrict ourselves to these dimensions. We thus introducethe diffuse Willmore functional

Wε(u) =

1c0

∫Ω

1ε

(ε∆u− 1

ε W′(u)

)2dx u ∈ X

+∞ else

and the Radon measures associated with a sequence uε with bounded energy Sε(uε) +Wε(uε)

µε(U) =1

c0

∫U

ε

2|∇uε|2 +

1

εW (uε) dx, αε(U) =

1

c0

∫U

1

εv2ε dx.

By Young’s inequality ε4a

2 + εb2 ≥ ab we can control the BV -norms of G(uε) =∫ uε−1

√2W (s) ds

and invoking the compactness theorems for BV -functions and Radon measures to see that there areu ∈ BVloc(Rn) and finite Radon measures µ, α with support in Ω such that, up to a subsequence,

uε → u strongly in L1(Ω), µε∗ µ, αε

∗ α, |Du| ≤ 2µ.

Clearly by construction u = χE−χEc for some E ⊂ Ω, and from the above argument is not difficultto see that E is a Caccioppoli set. In three dimensions, however, E may well be empty using theconstruction of [MR14], where two converging spheres are connected by a catenoid with boundedWillmore energy. A diagonal sequence shows that the same is possible with phase fields.

In [RS06] it has been shown that µ is in fact the mass measure of an integral n−1-varifold with|∂∗E| ≤ µ and H2

µ · µ ≤ α for generalised mean curvature Hµ of µ. Unlike µ, α can still behavewildly.

We aim to minimise the Willmore energy among connected surfaces with given area S. Toprescribe area µ(Ω) = S, we include a penalisation term of

1

εσ(Sε − S)2

in the energy functional.

2.2. Equipartition of Energy. The Cahn-Hilliard energy forces phase fields to be close to ±1 inphase and make transitions between the phases along layers of width proportional to ε. So while itmake sense that the terms ε

2 |∇uε|2 and 1

ε W (uε) should scale the same way to give us the surfacemeasure, it is by no means obvious that, for sequences uε of bounded Wε energy, they make anequal contribution to the measure µε. The difference of their contributions is controlled by thediscrepancy measures

ξε(U) =1

c0

∫U

ε

2|∇uε|2 −

1

εW (uε) dx,

their positive parts ξε,+(U) = 1c0

∫U

(ε2 |∇uε|

2 − 1ε W (uε)

)+

dx and their total variation measures

|ξε|(U) = 1c0

∫U

∣∣ ε2 |∇uε|

2 − 1ε W (uε)

∣∣ dx. We will see that for a suitable recovery sequence, |ξε|vanishes exponentially in ε; more generally the discrepancy goes to zero in general sequences uεalong which αε + µε stays bounded as shown in [RS06, Propositions 4.4, 4.9]. The result hasbeen improved along subsequences in [BM10, Theorem 4.6] to Lp-convergence for p < 3/2 andconvergence of Radon measures for the gradients of the discrepancy densities.

The control of these measures is extremely important since they appear in the derivative of thediffuse (n− 1)-densities r1−nµε(Br(x)).

2.3. Connectedness. In order to ensure that the support of the limiting measure µ is connectedwe include an auxiliary term Cε in the energy functional. To define it, we use an adapted geodesicdistance

dF (u)(x, y) = inf

∫K

F (u) dH1

∣∣∣∣K connected, x, y ∈ K,H1(K) ≤ ω(ε)

, ω(ε)→∞ as ε→ 0,


which is integrated against a suitable weight function. We thus take

Cε(u) =1

ε2

∫Ω

∫Ω

φ(u(x))φ(u(y)) dF (u)(x, y) dxdy,

where φ ∈ Cc(−1, 1) resembles a bump, i.e.,

φ ≥ 0, φ > 0 = (ρ1, ρ2) b (−1, 1),

∫ 1

−1

φ(u) du > 0

and F ∈ C0,1(R) is chosen to be zero where φ does not vanish:

F ≥ 0, F ≡ 0 on [ρ1, ρ2], F (−1), F (1) > 0.

The heuristic idea of this is that if the support of the limiting measure spt(µ) is connected, thenso should the set ρ1 < uε < ρ2. These level sets away from ±1 can be heuristically viewed asapproximations of spt(µ), and in other situations, they can be seen to Hausdorff-converge againstit, see [CC95]. This is not quite true in our situation (at least in three dimensions where we canget additional points in the Hausdorff limit), but the intuition still holds.

If φ(uε) > 0 = ρ1 < uε < ρ2 is connected, we can connect any two points x, y ∈ Ω such thatφ(uε(x))φ(uε(y)) > 0 with a curve of length zero, hence dF (uε)(x, y) = 0 and both the integrandand the double integral vanish.

If on the other hand spt(µ) is disconnected, then we expect that dF (uε) should be able toseparate different connected components such that lim infε→0 Cε(uε) > 0. The core part of ourproof is concerned with precisely that. We need to show that φ detects components of the interfaceand that dF (uε) separates them. For the first result, we need to understand the structure of theinterfaces converging to µ, for the second part, we need to exclude the possibility of phase fieldsbuilding tunnels between different connected components of spt(µ) which collapse away in the massmeasure but keep the distance close to zero along a sequence. This is easy in two dimensions where1 = n− 1 but hard in three dimensions where curves have codimension two.

2.4. Main Results. We have the following new results for phase-fields with bounded Willmoreenergy and perimeter.

Theorem 2.1. Let n = 2 and uε ∈ X a sequence such that

lim supε→0

(Wε + Sε) (uε) <∞.

Denote µ = limε→0 µε for a subsequence along which the limit exists and take any Ω′ b Rn\spt(µ).Then |uε| → 1 uniformly on Ω′.

For this and the following theorems, we take the continuous representative of uε which existssince W 2,2 −→ C0 in n = 2, 3 dimensions.

Theorem 2.2. Let n = 2 and D b (−1, 1). Then u−1ε (D)→ spt(µ) in the Hausdorff distance. If

n = 3, up to a subsequence, u−1ε (D) Hausdorff-converges to a set K which contains spt(µ).

For our application to connectedness, we define the total energy of an ε-phase field as

(2.1) Eε(u) =

Wε(u) + ε−σ (Sε(u)− S)2 + ε−κ Cε(u) u ∈ X+∞ else

for σ ∈ (0, 4), κ > 1. Using our previous results, we can show the following.

Remark 2.3. Existence of mininimisers for the functional Eε is a simple exercise in the directmethod of the calculus of variations, since uniform convergence of a minimising sequence for fixedε guarantees convergence of the distance term.


Theorem 2.4. Let n = 2, 3 and uε ∈ X a sequence such that lim infε→0 Eε(uε) < ∞. Then thediffuse mass measures µε converge weakly* to a measure µ with connected support spt(µ) ⊂ Ω andarea µ(Ω) = S.

Using [RS06], µ is also the mass measure of an integral varifold. The main result of [RS06] canbe applied to deduce Γ-convergence of our functionals in the following sense:

Corollary 2.5. Let n = 2, 3, S > 0, Ω b Rn and E b Ω, with smooth boundary ∂E ∈ C2 witharea Hn−1(∂E) = S. Then

Γ(L1(Ω))− limε→0Eε(χE − χEc) =

W(∂E) ∂E is connected

+∞ otherwise

Here we have adopted the notation of [RS06] where Γ-convergence is said to hold at a pointif the lim inf- and lim sup-inequalities hold at that point. This distinction is necessary since it isnot clear what Γ-convergence properties hold at other points u ∈ BV (Ω, −1, 1) since Wε doesnot converge to the L1-lower semi-continuous envelope of W at non-embedded sets even in twodimensions.

This issue stems from the fact that the stationary Allen-Cahn equation −∆u + W ′(u) = 0admits global saddle-solutions which have zeros along the coordinate axes and are positive in thefirst and third quadrants and negative in the second and fourth ones or even more degenerateones, see [dPKPW10]. These solutions can be used to approximate transversal crossings with zeroWillmore energy, while the lower semi-continuous envelope of Willmore’s energy/Euler’s elasticaenergy becomes infinite at those points. Interestingly enough, the crossing has zero Willmore-energy as a varifold, so that the energy can still be justified, despite the fact that it does not givethe sensible result in our situation.

While heuristic considerations suggest – and numerical simulations appear to confirm – that atleast in n = 2 dimensions and when 0 /∈ spt(φ), our additional term in the energy might preventsaddles, we do not investigate convergence at non-embedded points further.

Remark 2.6. Uniform convergence does generally fail in three dimensions. This can be seen bytaking a sequence uε as the optimal interface approximation which will be given in the proof ofCorollary 2.5 and adding to it a perturbation g((x − x0)/ε) for any g ∈ C∞c (Rn). It can easilybe verified that the energy will remain finite (at least if |g| 1) but clearly the functions do notconverge uniformly.

Heuristic arguments show that for modifications uε = uε+εα g(ε−β ·), the coefficients α = β = 1should be optimal in three dimensions and α = 1/2, β = 1 if n = 2, so we expect a convergencerate of

√ε in the two-dimensional case.

Remark 2.7. Since Sε(uε) is uniformly bounded, standard Modica-Mortola arguments show thatuε → χE − χΩ\E in Lp for p < 4/3. We are going to show in Lemma 3.1 that the sequenceuε is uniformly bounded in L∞(Rn) when additionally lim supε→0W(uε) < ∞, which impliesLp-convergence for all p <∞.

3. Proofs

The proofs are organised in the following way. First, anticipating the results of section 3.4,we show Γ-convergence of Eε to W. We decided to move the proof to the beginning, since it isvirtually independent of all the other proofs in this article and can be isolated, but introduces theoptimal interface transitions which will be needed later. After that, we proceed chronologicallywith technical Lemmata (sections 3.2 and 3.3) and the proof of Theorems 2.1, 2.2 and 2.4 in section3.4.


3.1. Proof of Γ-Convergence. We now proceed to prove Corollary 2.5.

Proof of the lim inf-inequality: It will follow from Theorem 2.4 that Eε(uε) → ∞ if ∂E is discon-nected. If ∂E is connected, the main part of this inequality is to show that if uε → χE − χEcin L1(Ω) and µε(Ω) ≤ S + 1, then lim infε→0 Eε(uε) ≥ W(∂E). Since Eε ≥ Wε and enforces thesurface area estimate, we obtain with [RS06] that

lim infε→0

Eε(uε) ≥ W(∂E).

Proof of the lim sup-inequality: We may restrict our analysis to the case of connected boundarieswith area Hn−1(∂E) = S. In this proof, we will construct a sequence uε ∈ X such that

limε→0||uε − (χE − χEc)||1,Ω = 0, lim

ε→0Eε(uε) =W(∂E).

The construction is standard and holds for arbitrary n ∈ N. We take the solution of the onedimensional problem and apply it to the signed distance function of ∂E, thus recreating the shapeof ∂E with an (approximate) optimal profile for the transition from −1 to 1. For δ > 0 consider

Uδ := x ∈ Rn | dist(x, ∂E) < δ.

Since E b Ω, Uδ ⊂ Ω for all sufficiently small δ, and since ∂E ∈ C2 is embedded, there is δ > 0such that

ψ : ∂E × (−δ, δ)→ Uδ, ψ(x, t) = x+ t νx

is a diffeomorphism. We denote the partial inverse to ψ which projects onto ∂E by π and byd(x) = sdist(x, ∂E) the signed distance of x from ∂E which is positive inside E and negativeoutside E, C2-smooth on Uδ and satisfies (see e.g. [GT01, Section 14.6])

∇d(x) = νπ(x), ∆d(x) = Hπ(x) + Cπ(x) · d(x) +O(d(x)2).

Let us consider the optimal transition between −1 and 1 in one dimension. This optimal profile isa stationary point of S1, i.e. a solution of −q′′ +W ′(q) = 0 (and thus a zero energy point of W1)with the side conditions that limt→±∞ q(t) = ±1. Note that the optimal profile satisfies

q′′ = W ′(q) ⇒ q′′ q′ = W ′(q) q′ ⇒ d

dt

((q′)2 −W (q)

)= 0,

so we find that (q′)2−W (q) ≡ c. We look for transitions where both (q′)2 and W (q) are integrableover the whole real line, and since limt→±∞W (q(t)) = 0, we see that c = 0 and (q′)2 ≡W (q). Thisgives us equipartition of energy already before integration. For simplicity, we focus on W (q) =

(q2 − 1)2/4 which has the optimal interface q(t) = tanh(t/√

2). Note that the functional rescalesappropriately under dilations of the parameter space so that we have equipartition of energy beforeintegration also in the ε-problem. The disadvantage of the hyperbolic tangent is that it makes thetransition between the roots of W only in infinite space, so we choose to work with approximationsqε ∈ C∞(R) such that

(1) qε(t) = q(t) for |t| ≤ δ/(3ε),(2) qε(t) = 1 for t ≥ δ/(2ε),(3) qε(−t) = −qε(t),(4) q′ε > 0.

Then we set

uε(x) = qε(d(x)/ε).

A direct calculation establishes that with this choice of uε, we have |Sε(uε)−S| ≤ εγ for all γ < 2,|ξε| ≤ εm for all m ∈ N and limε→0Wε(u

ε) =W(∂E). It remains to show that limε→0 ε−κCε(u

ε) =


0. We will show that even Cε(uε) ≡ 0 along this sequence. Since ∂E is connected and ψ is a

diffeomorphism, all the level sets

uε = ρ = ψ(∂E, q−1ε (ρ))

are connected manifolds for ρ ∈ (−1, 1). We know that

φ(uε) > 0 = ρ1 < uε < ρ2

and pick any ρ ∈ (ρ1, ρ2). Now let x, y ∈ Ω, φ(uε(x)), φ(uε(y)) > 0. We can construct a curvefrom x to y by setting piecewise

γ1 : [d(x), ρ]→ Ω , γ1(t) = π(x) + t νπ(x),

γ3 : [ρ, d(y)]→ Ω , γ3(t) = π(y) + t νπ(y)

and γ2 any curve connecting γ1(ρ) to γ3(ρ) in uε = ρ. This curve exists since connected manifoldsare path-connected. The curve γ = γ3 ⊕ γ2 ⊕ γ1 connects x and y and satisfies by constructionφ(γ(t)) > 0, so F (γ(t)) ≡ 0. Therefore we deduce

dF (uε)(x, y) = 0

if φ(uε(x)), φ(uε(y)) 6= 0, noting that the connecting curves have uniformly bounded length andω(ε)→∞. Thus in particular

1

ε2

∫Ω×Ω

φ(uε(x))φ(uε(y)) dF (uε)(x, y) dxdy ≡ 0.

3.2. Auxiliary Results I. A lot of our proofs will be inspired by [RS06] which again draws from[HT00]. We will generally cite [RS06] because it treats the more relevant case for our study. Inthis section, we will prove a number of auxiliary results which concern either general propertiesof phase fields or properties away from the support of the limiting measure µ which will enableus to investigate their convergence later. Results concerning phase interfaces are postponed untilsection 3.3. We start with a partial regularity lemma which is a slight improvement upon [RS06,Proposition 3.6].

Lemma 3.1. Let n = 2, 3, Ω b Rn and uε ∈ X such that α := lim supε→0Wε(uε) < ∞. Thenthere exists ε0 > 0 such that for all ε < ε0

(1) ||uε||∞,Rn ≤ C where C is a constant depending only on α and n.(2) uε is 1/2-Holder continuous on ε-balls, i.e.

|uε(x)− uε(y)| ≤ C√ε|x− y| 12 ∀ x ∈ Rn, y ∈ B(x, ε).

Again, the constant C depends only on α and n.

Proof. We will argue using Sobolev embeddings for blow ups of uε onto the natural length scale.In the first step, we show the set |uε| > 1 to be small. In the second step, we estimate theL2-norm of the blow ups, in the third we estimate the full W 2,2-norm and use suitable embeddingtheorems to conclude the proof of regularity.


Step 1. First observe that

αε(Ω) ≥ αε(uε > 1)

=1

c0

∫uε>1

1

ε

(ε∆uε −

1

εW ′(uε)

)2

dx

=−2

c0 ε

∫∂uε>1

W ′(uε) ∂νuε dHn−1

+1

c0

∫uε>1

ε (∆uε)2 +

2

εW ′′(uε) |∇uε|2 +

1

ε3(W ′(uε) )2 dx

≥ 1

c0

∫uε>1

ε (∆uε)2 +

4

ε|∇uε|2 +

1

ε3(W ′(uε) )2 dx

using that W ′′(t) ≥ 2 for t ≥ 1. The boundary integral vanishes since uε ∈ W 2,2 −→ C0,1/2 iscontinuous and W ′(1) = 0 if uε > 1 is of finite perimeter. If this is not the case, take θ 1converging from above such that uε > θ is of finite perimeter. This holds for almost all θ ∈ R.The sign of the boundary integral can be determined since ∂νuε < 0 on the boundary of uε > θand W ′(θ) > 0 for θ > 1 so that the same inequality can still be established. By symmetry, thesame argument works for uε < −1.

Step 2. Let xε ∈ Ω be an arbitrary sequence and define the blow up sequence uε : Rn → R by

uε(y) = uε(xε + εy).

Then we observe that∫B(0,2)

u2ε dx =

∫B(0,2)

( |uε| − 1 + 1)2

dy

≤∫B(0,2)

( (|uε| − 1)+ + 1)2

dy

≤ 2

∫B(0,2)

( |uε| − 1)2+ + 1 dy

≤ 2 ε3−n∫|uε|>1

1

ε3W ′(uε)

2 dy + 2n+1 ωn

≤ 2

2n ωn + c0 ε3−n αε(Ω)

.

As usual, ωn denotes the volume of the n-dimensional unit ball. In exactly the same way with aslightly simpler argument we obtain∫

B(0,2)

(W ′(uε))2 dy ≤ C(α, n).

Step 3. Now a direct calculation shows that∫B(0,2)

(∆uε −W ′(uε))2

dy =

∫B(0,2)

(ε2∆uε −W ′(uε))2 (xε + εy) dy

= c0ε3−n αε(B(xε, 2ε)).

Thus

||∆uε ||2,B(0,2) ≤ ||∆uε −W ′(uε) ||2,B(0,2) + ||W ′(uε) ||2,B(0,2)

≤√c0 ε3−n αε(Ω) +

√C(α, n).

In total, we see that|| uε ||2,B(0,2) + ||∆uε ||2,B(0,2) ≤ C(α, n)


for all 0 < ε < 1 so small that αε(Ω) ≤ α(Ω) +1. Using the elliptic estimate from [GT01, Theorem9.11], we see that

||uε||2,2,B(0,1) ≤ C(α, n),

where we absorbed the constant depending only on n and the radii into the big constant. Usingthe Sobolev embeddings

W 2,2(B(0, 1)) −→W 1,6(B(0, 1)) −→ C0,1/2(B(0, 1))

we deduce that

|uε|0,1/2,B(0,1) ≤ C(n, α),

again absorbing the embedding constants into the constant. In particular, this shows that

||uε||∞,B(0,1) ≤ C(n, α).

But since this holds for all sequences xε, we can deduce that

||uε ||∞,Rn ≤ C(n, α).

Furthermore, for x, z ∈ Rn with |x− z| < ε < ε0, we choose xε = x to deduce

|uε(x)− uε(y)| = |uε(0)− uε( (y − x)/ε)| ≤ C(n, α) | (y − x)/ε |12 =

C(n, α)√ε|x− y| 12 .

Remark 3.2. Without prescribing boundary conditions as in our modified space X, the resultcould still be salvaged on compactly contained subsets. Techniques for estimating quantities over|uε| > 1 in that case can be found in [RS06, Proposition 3.5], which we include below for thereaders’ convenience.

Proposition 3.3. [RS06, Proposition 3.5] For n = 2, 3, Ω ⊆ Rn, ε > 0, uε ∈ C2(Ω), vε ∈ C0(Ω),

−ε∆uε +1

εW ′(uε) = vε in Ω,

and Ω′ b Ω, 0 < r < dist(Ω′, ∂Ω), we have∫|uε|≥1∩Ω′

W ′(uε)2 ≤ Ck(1 + r−2kε2k)ε2

∫Ω

v2ε + Ckr

−2kε2k

∫|uε|≥1∩Ω

W ′(uε)2

for all k ∈ N0.

A useful rescaling property is the following observation from the proof of [RS06, Theorem 5.1].

Lemma 3.4. Let uε : B(x, r)→ R, λ > 0 and uε : B(0, r/λ)→ R with

uε(y) = uε(x+ λy).

Set r := r/λ, ε := ε/λ,

µε :=1

c0

(ε

2|∇uε|2 +

1

εW (uε)

)Ln, αε :=

1

c0 ε

(ε∆uε −

1

εW ′(uε)

)Ln.

Then

r1−nµε(B(0, r)) = r1−n µε(B(x, r)), r3−n αε(B(0, r)) = r3−n αε(B(x, r)).

The discrepancy measures ξε,± behave like µ under rescaling.

Proof. This can be seen by a simple calculation similar to the one in the proof of 3.1.

For the reader’s convenience we include the following classical monotonicity result.


Lemma 3.5. [RS06, Lemma 4.2] For x ∈ Rn we have

d

dρ

(ρ1−n µε(B(x, ρ))

)= −ξε(B(x, ρ))

ρn+

1

c0 ρn+1

∫∂B(x,ρ)

ε 〈y − x,∇uε〉2 dHn−1(y)

+1

c0 ρn

∫B(x,ρ)

vε 〈y − x,∇uε〉dy

In low dimensions n = 2, 3, the second and third term in the monotonicity formula can easilybe estimated after integration. While the result is known, we fixed minor details in the proof of[RS06, Proposition 4.5], so we include it here for completeness.

Lemma 3.6. [RS06, Proposition 4.5] Let 0 < r < R < ∞ if n = 3 and 0 < r < R ≤ 1 if n = 2,then

r1−nµε(B(x, r)) ≤ 3R1−nµε(B(x,R)) + 2

∫ R

r

ξε,+(B(x, ρ))

ρndρ

+1

2 (n− 1)2αε(B(x,R)) +

r3−n

(n− 1)2αε(B(x, r))

+R2

0 R1−n

(n− 1)2αε(B(x,R))(3.1)

where R0 := minR,RΩ and RΩ is a radius such that Ω ⊂ B(0, RΩ/2)

Proof. Without loss of generality we may assume that x = 0 and write Bρ := B(0, ρ), f(ρ) =ρ1−nµε(Bρ). Observe that for any function g : BR → R we have

∫ R

r

ρ−n∫Bρ

g(x) dxdρ =

∫BR

g(x)

∫ R

max|x|,rρ−n dρdx

=1

n− 1

∫BR

g(x)

(1

max|x|, rn−1− 1

Rn−1

)dx

and

∫ R

r

ρ−(n+1)

∫∂Bρ

g(x) dHn−1 dρ =

∫BR\Br

g(x)

|x|n+1dx.


Using this to integrate the derivative we obtain using Young’s inequality

f(R)− f(r) =

∫ R

r

f ′(ρ) dρ

=

∫ R

r

− ξε(Bρ)ρn

dρ+1

c0

∫BR\Br

ε 〈∇uε, y〉2

|y|n+1+

1

n− 1

vε 〈y,∇uε〉|y|n−1

dy

+1

(n− 1) c0 rn−1

∫Br

vε 〈y,∇uε〉dy −1

(n− 1) c0Rn−1

∫BR

vε 〈y,∇uε〉dy

≥∫ R

r

− ξε,+(Bρ)

ρndρ

+1

c0

∫BR\Br

ε 〈∇uε, y〉2

|y|n+1− 1

n− 1

((n− 1) ε

〈y,∇uε〉2

|y|2(n−1)+

1

4 (n− 1) εv2ε

)dy

− 1

c0 rn−1

∫Br

λε 〈y,∇uε〉2

2 |y|2+

1

2λ

|y|2 v2ε

(n− 1)2 εdy

− 1

c0Rn−1

∫BR

λε 〈y,∇uε〉2

2 |y|2+

1

2λ

|y|2 v2ε

(n− 1)2 εdy

≥∫ R

r

− ξε,+(Bρ)

ρndρ− 1

4 (n− 1)2

∫BR\Br

1

εv2ε dy

− λ f(r)− 1

2λ

r2

(n− 1)2 rn−1

∫Br

v2ε

εdy

− λ f(R)− 1

2λ

R20

(n− 1)2Rn−1

∫BR

v2ε

εdy.

where λ ∈ (0, 1). Here we used that n = 2, 3 to obtain that 2(n − 1) ≤ n + 1, so that |y|n+1 ≤|y|2(n−1) for all |y| if n = 3 and for |y| ≤ 1 if n = 2. When we bring all the relevant terms to theother side, this shows that

(1 + λ) f(R)− (1− λ) f(r) ≥ −∫ R

r

ξε,+(Bρ)

ρndρ− 1

4 (n− 1)2αε(BR \Br)

− r3−n

2λ (n− 1)2αε(Br)−

R20

2λ (n− 1)2Rn−1αε(BR).

Setting λ = 1/2 and multiplying by two proves the Lemma.

Remark 3.7. If n = 3, we may let R →∞ and subsequently ε→ 0, r → 0 and finally λ→ 0 thatwe have

lim supr→0

r1−nµ(B(x, r)) ≤ 1

4 (n− 1)2α(Ω)

at every point x ∈ R3 such that α(x) = 0 (i.e. when limr→0 α(Br) = 0). Using the results of[RS06], µ is an integral varifold, so this yields a Li-Yau-type [LY82] inequality

θ∗(µ, x) = lim supr→0

µ(B(x, r))

π r2≤ 1

16πα(Ω).

This inequality is usually found with a 4 in place of the 16 which stems from a different normali-sation of the mean curvature and W(µ) in the place of α, see also the proof of [Top98, Lemma 1]or [KS12, Proposition 2.1.1].


In n = 2 dimensions, we cannot do this since we had to assume R ≤ 1. Indeed, an inequalityof this type cannot hold since circles with large enough radii have arbitrarily small elastic energy.Still, setting R = 1, a similar bound on the multiplicity in terms of α and S can still be obtained.

The version we will use of the above inequality is the simplified expression

r1−nµε(B(x, r)) ≤ 3R1−nµε(B(x,R)) + 3αε(B(x,R)) + 2

∫ R

r

ξε,+(B(x, ρ))

ρndρ.

This holds generally if n = 3, and when R ≤ 1 if n = 2. Furthermore, we have the followingestimate for the positive part of the discrepancy measures.

Lemma 3.8. [RS06, Lemma 3.1] Let n = 2, 3. Then there are δ0 > 0,M ∈ N such that for all0 < δ ≤ δ0, 0 < ε ≤ ρ and

ρ0 := max2, 1 + δ−Mε ρwe have

ρ1−nξε,+(B(x, ρ)) ≤ C δ ρ1−n µε(B(x, 2ρ)) + C δ−Mε2 ρ1−n∫B(x,ρ0)

1

εv2ε dx

+ C δ−Mε2 ρ1−n∫B(x,ρ0)∩|uε|>1

1

ε3W ′(uε)

2 dx+C ε δ

ρ.

The following lemma is the key ingredient in order to obtain our required convergence results.In a slight abuse of notation, we will denote the functionals defined by the same formulas byWε, Sεagain, although they are given on spaces over B1 := B(0, 1), not Ω.

Lemma 3.9. Let n = 2, 3, θ ∈ (0, 1), 0 < η < 1/2. Consider the subsets

Y 2 := u ∈W 2,2(B1) : |u(0)| ≤ θ

for n = 2 dimensions and

Y 3ε :=

u ∈W 2,2(B1) : |u(0)| ≤ θ and αε(B√ε) +

∫B√ε∩|uε|>1

W ′(u)2

ε3dx ≤ εη

.

Define Fε : W 2,2(B1)→ [0,∞) as

Fε(u) =Wε(u) + Sε(u).

Then θ0 := lim infε→0 infu∈Y 2 Fε(u) > 0 if n = 2 and

θ0 := lim infε→0

infu∈Y 3

ε

Fε(u) > 0

if n = 3. The same works if instead u(0) ≥ 1/θ.

Proof: By Holder continuity, the condition that |u(0)| ≤ θ leads to the creation of an infinitesimaldiffuse mass density ε1−nµε(Bε) ≥ cn,α,θ. We will use the monotonicity formula to integrate thisup to show that if α = 0, macroscopic mass is created as well. In three dimensions, there is anadditional technical complication which forces us to make two steps, one from the ε-scale to thelength scale of

√ε and a second one to the original scale.

Step 1. In a first step we show that the diffuse mass densities are uniformly bounded on largeenough length scales. For x ∈ Ω, set fε(ρ) := ρ1−nµε(B(x, ρ)). Without loss of generality, we mayassume that fε(1) ≤ 1 for small enough ε > 0 and that αε(B3/4)+

∫B3/4∩|uε|>1

1ε3W

′(uε)2 dx ≤ 1

due to [RS06, Proposition 3.5] (included here as Proposition 3.3).


Take δ = log(ε)−2 in Lemma 3.8 to obtain from Lemma 3.6 that for 0 < r < R = 1 we have

fε(r) ≤ 3 fε(R) + 3αε(BR) + 2

∫ R

r

ξε,+(Bρ)

ρndρ

≤ Cα,n + C

∫ R

r

1

log(ε)2

fε(2ρ)

ρ+

ε

log(ε)2ρ2

+ ε2 log(ε)2Mρ−n

(αε(Bρ) +

∫Bρ∩|uε|>1

W ′(uε)2

ε3dx

)dρ

≤ Cα,n +C

log(ε)2

∫ R

r

fε(2ρ)

ρdρ+

Cε

log(ε)2

(1

r− 1

R

)+C ε2 log(ε)2M

n− 1

(r1−n −R1−n)

≤ Cα,n +C

log(ε)2

∫ 2R

r

fε(ρ)

ρdρ

for a uniform constant Cα,n for r ≥ ε if n = 2 and for r ≥√ε if n = 3. We use Gronwall’s

inequality backwards in time to deduce that

fε(r) ≤ Cα,n exp

(C

log(ε)2

∫ 2R

r

1

ρdρ

)≤ Cα,n

on [ε,∞) if n = 2 and on [√ε,∞) if n = 3.

Step 2. If n = 3 and additionally

αε(B√ε) +

∫B√ε∩|uε|>1

W ′(u)2

ε3dx ≤ εη,

we can estimate the terms in the second line more sharply to obtain uniform boundedness of fε(r)also for ε ≤ r ≤ R =

√ε.

Step 3. Now we turn to the proof of the statement. For a contradiction, assume that (αε +µε)(B1)→ 0 for a suitable sequence uε. The functions uε are C0,1/2-Holder continuous with Holderconstant C/

√ε for a uniform C ≥ 0 on B1/2, as can be obtained like in Lemma 3.1. The only

difference is that we need to use [RS06, Proposition 3.5] (included here as Proposition 3.3) toestimate

∫|uε|>1

1ε3W

′(uε)2 dx due to the lack of boundary values. It follows that u(x) ≤ 1+θ

2 for

x ∈ B(0, cε) for some uniform c > 0, which implies

ε1−nµε(Bε) ≥1

εn

∫B(0,cε)

W

(1 + θ

2

)dx =: cn,α,θ > 0.

In the following, we will assume n = 3. The two-dimensional case follows with an easier argumentof the same type. First we deduce that

fε(ε) ≤ 3 fε(√ε) + 3αε(B√ε)

+ C

∫ √εε

1

log(ε)2ρ+

ε

log(ε)2 ρ2+ ε2 log(ε)2Mρ−n εβ dρ.


Since the second line goes to zero as ε→ 0 and αε 0 by assumption, we deduce that fε(√ε) ≥

cn,α,θ/4 for all sufficiently small ε > 0. Finally, we obtain

fε(√ε) ≤ 3 fε(1) + 3αε(B1)

+ C

∫ 1

√ε

1

log(ε)2ρ+

ε

log(ε)2 ρ2+ ε2 log(ε)2Mρ−ndρ.

Again, the terms in the second line vanish with ε→ 0 and αε(B1)→ 0 by assumption, thus

fε(1) ≥ cn,α,θ16

for all sufficiently small ε > 0. But this contradicts the assumption that (αε + µε)(B1) → 0,so we are done. The case u(0) ≥ 1/θ follows similarly in two dimensions and by an immediatecontradiction to the definition of Y 3

ε in three dimensions.

The estimate above is not sharp, but suffices for our purposes. After this Lemma, everythingis in place to show uniform convergence of phase fields in in two dimensions in section 3.4, whilefurther results will be needed for Hausdorff convergence of the transition layers. In a very weakphrasing, Lemma 3.10 suffices, more precise versions (and the application to connectedness) needthe entire section 3.3. The reader mainly interested in the convergence of phase fields may skipahead now.

3.3. Auxiliary Results II. In this section, we will derive technical results concerning how phasefield approximations interact with the function φ as needed for the functional Cε to impose con-nectedness. While the previous section focused on estimates away from the interface, here weinvestigate the structure of transition layers close to spt(µ). The following Lemma is a special caseof [RS06, Proposition 3.4] with a closer attention to constants and the limit ε→ 0 already taken.

Lemma 3.10. Let x ∈ Rn, r, δ > 0, 0 < τ < 1− 1/√

2. Then

lim supε→0

µε(uε ≥ 1− τ ∩B(x, r)

)≤ 4 τ µ(B(x, r + δ)).

For all x ∈ Rn, there are only countably many radii r > 0 such that µ(∂B(x, r)) > 0. Thisfollows from the fact that µ is finite and that there are at most finitely many radii such thatµ(∂B(x, r)) ≥ 1/k, so that the union of those sets is countable. Thus for any r > 0 there ist ∈ (0, r) such that µ(∂B(x, t)) = 0. Letting δ → 0 at such a radius t (and using that thediscrepancy measures go to zero) gives us the following result (compare also [DMR14, Lemma 9]).

Corollary 3.11. For all x ∈ spt(µ), r > 0 and τ < 1/8 we have

lim infε→0

1

εLn ( |uε| < 1− τ ∩B(x, r)) > 0.

The following arguments rely more on the rectifiable structure of the measure µ that we areapproximating. Specifically, we introduce the diffuse normal direction by

νε :=∇uε|∇uε|

when ∇uε 6= 0 and 0 else. To work with varifolds, we introduce the Grassmannian G(n, n − 1)of n − 1-dimensional subspaces of Rn. We refer readers unfamiliar with varifolds or countablyrectifiable sets to the excellent source [Sim83]; an introduction with other focus which is easier tofind and covers most results relevant for us is [KP08]. Recall the following result.


Lemma 3.12. [RS06, Propositions 4.1, 5.1] Define the n− 1-varifold Vε := µε ⊗ νε by

Vε(f) =

∫Rn×G(n,n−1)

f(x, 〈νε〉⊥) dµε ∀ f ∈ Cc(Rn ×G(n, n− 1)).

Then there is an integral varifold V such that Vε → V as Radon measures on Rn × G(n, n − 1)(varifold convergence). The limit satisfies

µV = µ, H2µ µ ≤ α

where µV is the mass measure of V and Hµ denotes the generalised mean curvature of µ. Inparticular, W(µ) ≤ α.

The following result is a suitably adapted version of [RS06, Proposition 5.5] for our purposes.It shows that given small discrepancy measures and small oscillation of the gradient, a boundedenergy sequence looks very much like an optimal interface in small balls. Using our improvedbounds from Lemma 3.1, we can drop most of their technical assumptions.

Lemma 3.13. Let δ, τ > 0 and denote νε,n = 〈νε, en〉. Then there exist 0 < L <∞ depending onδ and τ only and γ > 0 depending on α, δ and τ such that the following holds for all x ∈ Rn. If

(1) |uε(x)| ≤ 1− τ and(2) |ξε|(B(x, 4Lε)) +

∫B(x,4Lε)

1− ν2ε,n dµε ≤ γ (4Lε)n−1

then also

• the blow up uε(y) = uε(x + εy) is C0,1/4-close to an optimal profile as introduced in theconstruction of the lim sup-inequality:

| ± uε − q(yn − t1)|0,1/4,B(0,3L) < δ

holds, where t1 := q−1(uε(x)).• |u(x, xn + t)| ≥ 1 − τ/2 for all Lε ≤ |t| ≤ 3Lε, where x = (x1, . . . , xn−1) and u changes

sign in between.

Proof. Without loss of generality, we may assume that x = 0 and write Br := B(0, r). Recall that

q′(t) =√

2W (q(t)) and limt→±∞ q(t) = ±1. Thus we can pick L > 0 such that |q(t)| ≥ 1 − τ/4for all t > L.

Assume for a contradiction that there is no constant γ > 0 such that the results of the Lemmahold. Then for γj → 0, there must be a sequence ujε such that |ujε(0)| ≤ 1− τ , Wε(u

jε) ≤ α+ 1 and

|ξjε |(B4Lε) +

∫B4Lε

1− ν2ε,n dµε ≤ γj (4Lε)n−1,

but the conclusions of the Lemma do not hold. Considering the blow ups uj : B4L → R withuj(y) = ujε(εy) we obtain

||uj ||2,2,B3L≤ Cα,n,L

like in Lemma 3.1, hence there is u ∈W 2,2(B3L) such that

uj u in W 2,2(B3L).

Since W 2,2 embeds compactly into W 1,2 and L4, we see that∫B3L

∣∣ |∇u|2/2−W (u)∣∣ dx = lim

j→∞

∫B3L

∣∣ |∇uj |2/2−W (uj)∣∣ dx

≤ limj→∞

ε1−nξjε(B4Lε)

≤ lim infj→∞

(4L)n−1γj

= 0


and when we set ∇u = (∂1u, . . . , ∂n−1u), we get∫B3L

|∇u|dx = limj→∞

∫B3L

|∇uj |dx

= limj→∞

∫B3L

√|∇uj |2 − |∂nuj |2 dx

≤ lim infj→∞

∫B4L

|∇uj |√

1−(νjn)2

dx

≤ lim infj→∞

(ωn (4L)n)1/2

(∫B4L

|∇uj |2(1− (νjn)2

)dx

) 12

≤ lim infj→∞

√8Lωn

(ε1−n

∫B4Lε

1− (νjn)2 dµε

) 12

≤ lim infj→∞

√8Lωn γj

= 0.

Thus we can see that

|∇u|2 = 2W (u), ∇u = (0, . . . , 0, ∂nu).

Clearly, this means that u(y) = p(yn) for a function p with p′ = ±√

2W (p) . Using that |u(0)| ≤1 − τ and the Picard-Lindeloff theorem on the uniqueness of the solutions to ODEs, we see thatp(yn) = ±q(yn − y) for some y ∈ R which can easily be fixed by the initial condition for p(0).

Since weak W 2,2-convergence implies strong C0,1/4-convergence in n = 2, 3 dimensions, we seethat there is j ∈ N such that the claim of the Lemma holds for ujε contradicting our assumption.Thus the Lemma is proven.

To deal with the rectifiable sets in the next section more easily we prove a structure result forrectifiable sets. The result seems standard, but we have been unable to find a reference for it. Asusual, we call a function on a closed set differentiable if it admits a differentiable extension to alarger open set.

Lemma 3.14. Let M be a countably k-rectifiable set in Rn. Denote by B the closed unit ball in kdimensions. Then there exist injective C1-functions fi : B → Rn with ∇fi 6= 0 on B such that

Hk(M \

∞⋃i=1

fi(B)

)= 0

and such that fi(B) ∩ fj(B) = ∅ for all i 6= j.

Proof. According to [KP08, Lemma 5.4.2] or [Sim83, Lemma 11.1] there is a countable collectionof C1-maps gi : Rk → Rn such that

M ⊂ N ∪∞⋃i=1

gi(Rk)

where Hk(N) = 0. Without loss of generality, N is assumed to be disjoint from the other sets.First we need to make the individual maps gi one-to-one. To do that, we define the set whereinjectivity fails in a bad way:

Ai :=x ∈ Rk | ∀ r > 0 ∃ y ∈ B(x, r) such that gi(x) = gi(y)

.


Due to the failure of local injectivity, we see that the Jacobian Jgi(x) vanishes on Ai. Since gi isa C1-function, the set Di := J−1

gi (0) is closed and by the Morse-Sard Lemma [Fed69, 3.4.3] then

Hk (gi(Di)) = 0.

Set Ui := Rk \Di. Now as in [EG92, Chapter 1.5, Corollary 2] we can use Vitali’s covering theorem

[EG92, Chapter 1.5, Theorem 1] to obtain a countable selection of closed balls Bji such that fi is

injective with non-vanishing gradient on Bji for all j ∈ N and

LkUi \ ∞⋃

j=1

Bji

= 0.

Since the boundary of a k-ball has Hausdorff dimension k − 1, we could equally well take openballs. Since C1-functions map sets of Lk-measure zero to sets of Hk-measure zero, we have shownthat we can write

M ⊂ N ∪∞⋃j=1

gi(B)

where Hk(N) = 0, gi : B → Rn is one-to-one, C1, and has a non-vanishing gradient everywhereon the closed ball B. The functions gm are obtained by rescaling suitable restrictions of gi fromBji to the unit ball. Finally, we have to cut out the sets that get hit by more than one functiongm. Inductively, we define

Um := B \ g−1m

(m−1⋃l=1

gl(B)

).

Finally, we use Vitali’s Lemma again to pick collections of closed balls Blm such that

Lk(Um \

∞⋃l=1

Blm

)= 0.

Rescaling the restricted functions from these balls and translating to the unit ball gives us theresult.

The proof of the following Lemma strongly resembles that of the integrality of µ in [RS06, Lemma4.2]. It is technically easier because we do not need the multi-layeredness of the approximatinginterfaces, but entails different complications because blow up to the tangent space is not possiblehere. Instead we use the preceding Lemma for a similar result on local C1-flatness.

Lemma 3.15. Let φ ∈ C0(R) such that φ ≥ 0 and∫ 1

−1φ(u) du > 0. If x ∈ spt(µ), then

lim infε→0

1

ε

∫B(x,r)

φ(uε) dx > 0

for all r > 0.

Proof. Step 1. As usual, we assume that x = 0, µ(∂B(x, r/2)) = 0 and denote B = B(x, r/2).This means that all the ε-balls of positive integral we are going to find will actually lie in B(x, r)and is a purely technical condition. Let ζ be a small constant to be specified later. For furtheruse, denote by B the closed unit ball in Rn−1.

As µ is an integral varifold, we know that spt(µ) is rectifiable. This means that there are

countably many C1-functions fi : B → Rn like in Lemma 3.14 such that

spt(µ) ⊂M0 ∪∞⋃i=1

fi(B), Hn−1(M0) = 0 fi(B) ∩ fj(B) = ∅


for i 6= j. Since µ has second integrable mean curvature µbH2µ ≤ α, we can further use the Li-Yau

inequality from Remark 3.7 to bound the maximum multiplicity of µ uniformly by

θmax ≤α(Ω)

16π,

at least Hn−1-almost everywhere. Now since Hn−1(spt(µ)) < +∞ we can find N ∈ N such that

Hn−1

((spt(µ) ∩B

)\N⋃i=1

fi(B)

)<

ζ

θmax.

Since fi is injective and has non-vanishing tangent maps everywhere, M :=⋃Ni=1 fi

(B)

is a

C1-manifold. We observe that

Hn−1(spt(µ) ∩B \M) <ζ

θmax

and hence

µ (B \M) < ζ.

Since the maps in question are smooth and the unit discs are orientable, for every i we can picka continuous unit normal field to fi(B) (e.g. using cross products). Since the discs are compactand disjoint (thus a positive distance apart), the fields defined on each disc separately induce acontinuous unit vector field on the union of their closures.

Now we use the Tietze-Urysohn extension theorem to obtain a vector field X on B such thatX = νM on M and projecting on the unit ball we ensure |X| ≤ 1. After an easy modification, wemay assume that |X| = 1 on a neighbourhood of M . We then define

G : Rn ×G(n, n− 1)→ R, G(x, S) = 〈Xx, νS〉2

where νS is one of the unit normals to S. Note that G is continuous since X is. Using the non-negativity of G and the fact that Txµ = TxM for Hn−1-almost every x ∈M ∩ spt(µ) we interpretµ as dual to C0(Rn ×G(n, n− 1)) and observe

〈µ,G〉 =

∫spt(µ)

θ(x)G(x, Txµ) dHn−1

≥∫

spt(µ)∩Mθ(x)G(x, Txµ) dHn−1

≥∫

spt(µ)∩Mθ(x)G(x, TxM) dHn−1

=

∫spt(µ)∩M

θ(x) dHn−1

= µ(M)

≥ µ(B)− ζ.

Step 2. By varifold convergence, we know that limε→0〈µε, G〉 = 〈µ,G〉 ≥ µ(B)−ζ, and |X|, |νε| ≤1 so

lim supε→0

∫B

∣∣1− 〈νε, X〉2∣∣dµε = lim supε→0

∫B

1− 〈νε, X〉2 dµε

≤ lim supε→0

(µε(B)− 〈µε, G〉)

≤ ζ.


For γ, ε, L > 0 we define the set

Uε,γ,L :=

x ∈ B

∣∣∣∣ 1

(4Lε)n−1

∫B(x,4Lε)

∣∣1− 〈νε, X〉2 ∣∣dµε > γ/4

.

Let x1, . . . , xK be points in Uε,γ,L being maximal for the property that the balls B(xi, 4Lε) aredisjoint. Then by definition

ζ ≥∫B

∣∣1− 〈νε, X〉2∣∣dµε ≥ K∑i=1

∫B(xi,4Lε)

∣∣1− 〈νε, X〉2∣∣ dµε ≥ K (4Lε)n−1 γ/4.

At the same time, we know that the balls B(xi, 8Lε) cover Uε,γ,L because otherwise we could bringin more disjoint balls, therefore

Ln(Uε,γ,L)

ε≤ K ωn (8Lε)n

ε

≤ 4 ζ

γ (4Lε)n−1

ωn (8Lε)n

ε

= 2n+4 ωn Lζ/γ.

For a given γ, we choose ζ = ζ(γ) such that this is ≤ µ(B)/4.Step 3. Knowing that |ξε|(B) → 0, we can use the same argument as in the second step to

show for

Vε,γ,L :=

x ∈ B

∣∣∣∣ |ξε| (B(x, 4Lε))

(4Lε)n−1> γ/2

the estimate

Ln(Vε,γ,L)

ε≤ µ(B)/4

for all sufficiently small ε > 0.Step 4. Now choose U such that U is a neighbourhood of M on which |X| = 1 and τ > 0 like

in Corollary 3.11 satisfying

lim infε→0

µε (U ∩ |uε| ≤ 1− τ) ≥ 3µ(B)

4.

This is easily achieved when µ(M) > 3µ(B)/4. Furthermore we take δ 1 suitably small forsmall deviations of the optimal interface to behave similarly enough, L and γ as in Lemma 3.13and ζ = ζ(γ). Using steps one through three, we see that

lim infε→0

Ln (|uε| ≤ 1− τ ∩ U \ (Uε,γ,L ∪ Vε,γ,L))

ε

≥ lim infε→0

Ln (|uε| ≤ 1− τ ∩ U)

ε− L

n (Uε,γ,L)

ε− L

n (Vε,γ,L)

ε≥ 3µ(B) /4− µ(B)/4− µ(B)/4

= µ(B)/4.

Using the argument of step 2, but this time in reverse, we can see that there are at least K pointsx1, . . . , xK in |uε| ≤ 1− τ ∩U \ (Uε,γ,L ∪ Vε,γ,L) such that the balls B(xi, 4Lε) are disjoint with

K ≥ µ(B)

8n+1 Ln εn−1.


Step 5. To apply Lemma 3.13, we must “freeze” the coefficients of the vector field X to a singleunit vector. We compute

1

(4Lε)n−1

∣∣∣∣ ∫B(xi,4Lε)

(1− 〈νε, X〉2

)−(1− 〈νε, Xi〉2

)dµε

∣∣∣∣=

1

(4Lε)n−1

∣∣∣∣ ∫B(xi,4Lε)

〈νε, Xi〉2 − 〈νε, X〉2 dµε∣∣∣∣

=1

(4Lε)n−1

∣∣∣∣ ∫B(xi,4Lε)

〈νε, Xi −X〉〈νε, Xi +X〉 dµε∣∣∣∣

≤ |Xi +X|C0(B(xi,4Lε)) · |Xi −X|C0(B(xi,4Lε))1

(4Lε)n−1·∫B(xi,4Lε)

dµε

≤ 2Cα,L,n |Xi −X|C0(B(xi,4Lε))

for all Xi such that |Xi| ≤ 1. When we set Xi = X(xi), the last term converges to zero - soeventually it is smaller than γ/4 and

1

(4Lε)n−1

∫B(xi,4Lε)

1− 〈νε, Xi〉2 dµε < γ/2.

Since xi ∈ U , we finally see that |Xi| = 1 and Lemma 3.13 can be applied.Step 6. Since uε is C0,1/4-close to a one-dimensional optimal profile on B(xi, 3Lε) which

transitions from −1 to 1, we see that for each s ∈ (−(1 − τ), (1 − τ)) there must be a pointyi ∈ B(xi, 3Lε) such that uε(yi) = s. By Holder continuity, we deduce that∫

B(xi,3Lε)

φ(uε) dx ≥ θ εn

for a constant θ depending on the support of φ and on α, n for the Holder constant. Since the ballsare disjoint by construction, we can add this up to

1

ε

∫B

φ(uε) dx ≥1

ε

M∑j=1

∫B(xi,3Lε)

φ(uε) dx

≥ 1

εM θ εn

≥ µ(B) θ

8n+1 Ln

> 0.

This concludes the proof.

We need one final result from geometric measure theory before we move on to our main results.The Lemma is standard knowledge for smooth manifolds with a vastly different proof.

Lemma 3.16. There is a universal constant C > 0 such that for all integral varifolds V with finiteWillmore energy and compact connected support we have

W(V ) ≥ C if n = 3 and W(V ) ≥ C

|V |(R2)if n = 2.

Proof. For a contradiction, assume that there is a sequence of varifolds Vi in Rn for n = 2, 3 suchthat |Vi|(Rn) = 1 and W(Vi) ≤ 1/i. We observe that connectedness already implies diam(Vi) ≤


|Vi|(Rn) = 1 for n = 2. If n = 3, we can use techniques from [Sim93] to bound the diameter ofspt(V ) in terms of its Willmore energy by

diam(V ) ≤ 2

π

√W(V ) · |V |(R3) ,

see also [Top98, Lemma 1] for a derivation of this inequality and the Li-Yau inequality. Thearguments are presented in the context of immersions but can easily be applied to integral varifoldswith connected support. Thus we know that up to translation we can assume spt(Vi) ⊂ B(0, 1),at least for large enough i. Then we may invoke Allard’s compactness theorem to find that thereis an integral varifold V such that Vi V and

• spt(V ) ⊂ B(0, 1),• |V |(Rn) = 1, and• W(V ) = 0.

But this contradicts the result of [MR14, Theorem 1] which shows that

W(V ) ≥ cn |V |(Rn) > 0.

The proof is presented there for n = 3, but can easily be adapted n = 2 as well.

For n = 3, another approach would be to use the Li-Yau inequality, which would give us anexplicit constant C = 16π. We chose this proof instead because it works for both n = 2 and n = 3the same way.

3.4. Proof of the Main Results. Having dealt with the necessary auxiliary results, we canproceed to prove our main results. We use the terminology of Lemma 3.9.

Proof of Theorem 2.1. Let Ω′ b Rn \ spt(µ) and τ > 0. Assume there is a sequence xε in Ω′ suchthat |uε(xε)| ≤ 1 − τ or |uε(xε)| ≥ 1 + τ . By definition, xε ∈ Ω and using compactness, there isx ∈ Ω′ ∩ Ω such that xε → x.

Let r > 0 such that B(x, 3r) ⊂ Rn \ spt(µ). Due to convergence, B(xε, r) ⊂ B(x, 2r) for allsufficiently small ε > 0. If we use the rescaling property from Lemma 3.4 and the minimisationproperty of Lemma 3.9 for n = 2, we see that

α(B(x, 2r)) ≥ 1

rlim infε→0

(r αε(B(xε, r)) + r−1 µε(B(xε, r))

)≥ 1

rlim infε→0

infu∈Y 2

Fε(u)

≥ θ0/r

with ε = ε/r as in Lemma 3.4. Letting r → 0, we obtain a contradiction.

If n = 3, neither the rescaling property of Lemma 3.4 nor the minimisation property of Lemma3.9 hold in as strong formulations. As we sketched in Remark 2.6, uniform convergence is, in fact,false. It is still an open question in which sense phase fields converge away from the interface inthree dimensions.

Proof of Theorem 2.2: We assume that uε → u strongly in L1(Ω) and that µε µ for our sequenceor a suitable subsequence (not relabelled). Let D b (−1, 1), then u−1

ε (D) b Ω. We can consider

u−1ε (D) instead without changing the Hausdorff limit to conform with standard approaches. By

the usual compactness results (see e.g. [KP08, Theorem 1.6.6]), there is a compact set K such thata further subsequence of u−1

ε (D) converges to K in Hausdorff distance. K can be computed as theKuratowski lower limit

K = x ∈ Rn | ∃ xε ∈ u−1ε (D) such that xε → x.


Step 1. We first show that K ⊂ spt(µ) if n = 2. Assume x ∈ K \ spt(µ). Then there existsr > 0 such that B(x, r) b R2 \ spt(µ). This means already that |uε| → 1 uniformly on B(x, r)showing that u−1

ε (D) ∩B(x, r) = ∅ for small enough ε, contradicting our assumption.Step 2. Now we prove that spt(µ) ⊂ K in n = 2, 3 dimensions. It suffices to show that for

x ∈ spt(µ), s ∈ D and r > 0 we have u−1ε (s)∩B(x, r) 6= ∅ for all sufficiently small ε, which implies

Hausdorff convergence.This is an easier version of the proof of Lemma 3.15. Again, we use Corollary 3.11 to show that

we have a point y ∈ B(x, r/2) to use Lemma 3.13 on and then use Lemma 3.13 to see that we getpoints in the pre-image of s close to y since uε is C0-close to an optimal interface in B(y, 3Lε).

If n = 2, the uniqueness of the limit also shows convergence for the whole sequence.

Finally, we show that Cε enforces connectedness.

Proof of Theorem 2.4: Let uε be a sequence such that Eε(uε) is bounded. Then in particular|µε(Ω) − S| ≤ εσ/2, so µε(Rn) = µε(Ω) is bounded and µε µ for some Radon measure µ – forthis and other properties see [EG92, Chapter 1]. Clearly

µ(Ω) ≥ lim supε→0

µε(Ω) = S

and on the other hand

µ(Ω) ≤ µ(Rn) ≤ lim infε→0

µε(Rn) = S

so µ(Ω) = S. If U = Rn \ Ω, we have

µ(U) ≤ lim infε→0

µε(U) = 0,

so spt(µ) =⋂U open,µ(U)=0 U

c ⊂ Ω. It remains to show that spt(µ) is connected. For a contra-

diction, assume that spt(µ) has at least two components. Since components are relatively closed,they are also compact. As spt(µ) is closed and contained in Ω, it is also compact. Since everyconnected component of spt(µ) by itself induces an integral varifold, Lemma 3.16 shows that thereare only finitely many connected components of spt(µ). This means that also the complement ofa component is relatively closed, hence compact.

Let C1 be a connected component of spt(µ) and set C2 := spt(µ)\C1. Since C1, C2 are compactand disjoint, we see that

δ := dist(C1, C2) = minx∈C1,y∈C2

|x− y| > 0.

We define the disjoint open sets

U1 := x ∈ Rn | dist(x,C1) < δ/3, U2 := x ∈ Rn | dist(x,C2) < δ/3.

Then we can consider two cases.Case 1. We assume that there is β < κ such that lim infε→0 ε

−βdistF (uε)(U1, U2) > 0. Then

lim infε→0

ε−βCε(uε) ≥ lim infε→0

∫U1

φ(uε(x)) dx · lim infε→0

∫U2

φ(uε(y)) dy

· lim infε→0

ε−β distF (uε)(U1, U2)

> 0

by our assumption and Lemma 3.15. Thus

lim infε→0

ε−κ Cε(uε) =∞,

which clearly contradicts our original assumption.


Case 2. We assume that 1 < β < κ and that lim infε→0 ε−βdistF (uε)(U1, U2) = 0. Then at

least for a suitable subsequence of ε → 0 (not relabeled) we have points xε ∈ ∂U1, yε ∈ ∂U2 andconnected sets Kε such that

xε, yε ∈ Kε, ε−β∫Kε

F (uε) dH1 ≤ 1.

Clearly

H1(F (uε) ≥ 1 ∩Kε) ≤ εβ .Without loss of generality we may assume that Kε ∩ (U1 ∪ U2) = ∅, otherwise take a connectedcomponent of Kε \ (U1 ∪ U2) meeting both sets. For the purpose of this proof we will make a fewsimplifying assumptions, namely that

φ > 0 = (ρ1, ρ2), −1 < θ1 < ρ1 < ρ2 < θ2 < 1, F ≥ 2 outside (θ1, θ2).

This allows us to neglect at least some constants and obviously has no effect on the structure ofthe functional. Now assume that there is zε ∈ Kε such that uε(zε) does not lie in [θ1, θ2]. Since Fis continuous and uε is Holder continuous on ε-balls, there is a constant c > 0 such that

F (uε(z)) ≥ 1 ∀ z ∈ B(zε, cε).

But since Kε is connected and not contained in B(zε, ε), it must contain a point wε ∈ ∂B(zε, cε),otherwise we directly reach a contradiction. Furthermore, the connected component Lε of Kε ∩B(zε, ε) containing zε and wε projects to f(Lε) = [0, cε) under the one-Lipschitz map f(z) =|z − zε|. Thus

H1(Lε) ≥ H1(f(Lε)) ≥ cε.On the other hand, by construction Lε ⊂ Kε ∩ F (uε) ≥ 1), so

cε ≤ H1(Lε) ≤ εβ .

For all sufficiently small ε > 0, this is a contradiction (as β > 1), and we see that uε ∈ [θ1, θ2] onKε. If n = 2, this contradicts uniform convergence and the proof is finished. If n = 3, we need afurther argument.

Take some γ, η > 0 to be fixed later. For fixed ε > 0, without loss of generality, we may assumethat xε = 0 and yε = rεe1 for rε ≥ δ/3. Denote by π1 the projection on the x1-coordinate. SinceKε is connected, we observe that [0, δ/3] ⊂ π1(Kε). Thus by construction, there are Nε ≥ δ/(6 εγ)points xiε ∈ Kε ∩ π−1

1 (2i · εγ) such that the balls B(xiε, εγ) are disjoint. Assume that there are

Mε > 0 balls among these such that

αε(B(xiε, εγ)) +

∫B(xiε,ε

γ)∩|uε|>1

W ′(uε)2

ε3dx ≥ εη .

Since the balls are disjoint and referring back to the first step in the proof of Lemma 3.1, thisimplies

2αε(Rn) ≥Mε εη,

so Mε ≤ 2 (α+ 1)/εη for all small ε. We deduce that

Mε

Nε≤ 2 (α+ 1)/εη

δ/(6 εγ)=

12 (α+ 1)

δεγ−η.

When we fix γ = 1/2, and 0 < η < γ we see that Mε/Nε → 0 so eventually for(1− Mε

Nε

)Nε ∼ Nε


balls we have αε(B(xiε, εγ))+

∫B(xiε,ε

γ)W ′(uε)

2

ε3 dx < εη. Denote byAε the collection of the centres of

these balls. As xiε ∈ Kε ⊂ Rn\(U1∪U2), for every 0 < r < δ/6 we know that B(xiε, 2r) ⊂ Rn\spt(µ)and

αε(B(xiε, εγ)) +

∫B(xiε,ε

γ)∩|uε|>1

W ′(uε)2

ε3dx < εη.

When we consider the rescaling uε : B(0, 1)→ R, uε(y) = uε(xiε + ry) for some sequence xiε ∈ Aε

and denote ε = ε/r we obtain (in n = 3 dimensions) that

αε(B(0, εγrγ−1)) +

∫B(0,εγ rγ−1)∩|uε|>1

W (uε)

ε3dx ≤ rη εη.

Without loss of generality, we can take r ≤ 1 and since we fixed γ ∈ (0, 1), this implies

αε(B(0, εγ)) +

∫B(0,εγ)∩|uε|>1

W (uε)

ε3dx ≤ εη.

Also, by construction µε 0 weakly as a Radon measure on B(0, 1). With the same argument thatgave us the lower bound on Nε, for any s ∈ (0, δ) we can obtain a sequence of points xε = xiεε ∈ Aεsuch that xε → x for some x ∈ Rn with π1(x) = s. In particular, this gives us infinitely many suchlimit points. We will show that each of them is an atom of α with a certain minimal size, whichobviously gives us a contradiction. By the weak convergence of Radon measures,

α(B(x, 2r)) ≥ lim supε→0

αε(B(x, 2r)) ≥ lim supε→0

αε(B(xε, r))

since for small enough ε > 0 we have B(xε, r) ⊂ B(x, 2r). But since µε 0 we have

lim supε→0

αε(B(xε, r)) = lim supε→0

(αε + µε)(B(0, 1)) ≥ θα,n,θ1,θ2

using Lemma 3.9. Letting r → 0 shows that in fact α(x) ≥ θα,n,θ1,θ2 for uncountably manypoints, leading to the contradiction we were looking for.

4. Computer Implementation

In a computer simulation, we try to find local minimisers of the ε-problem in n = 2 dimensionsby following a finite element implementation of the time-normalised L2-gradient flow of Eε withε = 1.5 ·10−2, κ = 1, σ = 2. The pictures below are obtained using roughly 250.000 H2-conformingbasis functions and a time-step of ε · 10−5. The distance function dF (uε) and its gradient areimplemented via Dijkstra’s algorithm in a fashion similar to the one of [BCPS10]. We note thatω(ε) can be chosen so large that it does not pose a restriction for the algorithm computing thedistance on our grid. The initial condition is the same for all simulations and can be seen in figure 1on the left; the domain is the disc of radius 1. There was no penalisation of the discrepancy measurein our simulations.

Note that the phase field in this figure is already relaxed by running the gradient flow approx-imately up to time t = 7.5 · 10−5, so that a smooth transition layer could form from the simplesharp “true” initial condition.

For practical purposes, we use two functions φ1, φ2 with support close to 1 and −1, respectively,rather than just one φ. Quite obviously, our proofs easily extend to that situation. By keepinglevel sets close to the edges connected, we create barriers that prevent the interface from splittingapart early in the process. The implementation will be described in greater detail in a forth-comingarticle [DW15].

We see in figure 1 that without the inclusion of the topological term, the transition layer disin-tegrates into several connected components along the gradient flow of Wε + ε−σ (Sε − S)2.


Figure 1. Gradient flow of Wε + ε−σ (Sε−S)2. From left to right: phase field ufor approximately t = 7.5 · 10−5, t = 3 · 10−4, t = 7.5 · 10−4 and t = 1.8 · 10−3.

To compare implementations of topological side conditions, we include the topological termsuggested in [DMR11], which penalises a deviation of a diffuse signed curvature integral from 2πin the simulation. This term prevents the initial pinch-off, but at a later time, the interface willpinch off in a more complicated way which keeps the diffuse winding number close to 2π. Thetrick is to pinch off simultaneously at several points as seen in figure 2. The far right plot infigure 2 illustrates the diffuse curvature density as distributed along the curve at pinch off time.We can observe the formation of a circle with negative total curvature ≈ −2π (due to the phasefield switching in the other direction from +1 to −1), and two components with total curvature≈ 2π so that the total curvature of the whole interface stays close to 2π.

Figure 2. Gradient flow with penalty on a diffuse winding number as suggestedin [DMR11]. From left to right: phase field u for approximately t = 3 · 10−4,t = 7.5 · 10−4 and t = 1.8 · 10−3, then a plot of the diffuse winding number densitydenoted T at time t = 1.8 · 10−3.

In figure 3, a flow for Eε with the additional term of Cε on the other hand can be seen tostably flow past those singular situations. The three left plots both here and in 2 correspondapproximately to the same times in the simulation.

Comparing the three scenarios above, we observe that there is virtually no difference in theplots at time 3 · 10−4 and that the plots for both modified (penalised using either the old or thenew method) functionals at time 7.5 · 10−4 still look very similar. It can thus be argued that thetopological condition does not affect the shape of the curve in a major way except when it has toin order to prevent loss of connectedness.

In figure 4, we see non-trivial geometric changes along the gradient flow for later times. Thisdemonstrates the necessity of continuing the flow beyond the critical times.


Figure 3. Evolution including our new topological penalty term Cε. From left toright: phase field u for approximately t = 3 ·10−4, t = 7.5 ·10−4 and t = 1.8 ·10−3,then a plot of the diffuse Willmore energy density (denoted W here) of the initialcondition.

Figure 4. Evolution including our new topological penalty term Cε for longtimes. From left to right: phase field u and diffuse Willmore energy density(denoted W here) first for approximately t = 6.6 ·10−3 and then for approximatelyt = 3.6 · 10−2.

It should be emphasised that our focus is not on implementing an approximate Willmore flowusing phase fields but on finding minimisers of the diffuse interface problem using a gradient flow.Existence of Willmore flow for long time and topological changes along it are still an open field ofresearch.

5. Conclusions

In this paper, we have developed a strategy to enforce connectedness of diffuse interfaces. Thestrategy fares well in applications and can efficiently be implemented and seems to be more generallyapplicable to a wider class of problems. We claim that our results can be extended to the followingsituations.

• We can include a hard volume constraint, for example

1

ε

(1

2

∫Ω

uε + 1 dx− V)2

for 0 < V < minLn(Ω), cn Sn/(n−1) or a soft volume constraint like

F

(1

2

∫Ω

uε + 1 dx

)


for continuous functions F ≥ 0. Here cn is the constant from the iso-perimetric inequalityin n dimensions.• Another popular constraint compatible with our functional and results is minimising a

distance from a given configuration as

Aε(u) =

∫Ω

|u− g|dλ

where λ is a finite Radon measure on Ω and g ∈ L1(Ω). This functional originates inproblems in image segmentation, but in our context it can be understood as prescribingcertain points to lie inside or outside the membrane according to experimental data; acomputationally stable choice would be for example

g =

1 Ω1

0 Ω2

−1 Ω3

, λ = LnbΩ.

This urges the functional to have Ω1 ⊂ E and Ω3 ⊂ Ec without preference on Ω2; theenergetic drive to form transitions on Ω2 is compensated by the volume constraint anddisappears as ε → 0. As for the volume contribution, the constraint can be included in ahard or soft penalisation.

In a soft penalisation, it suffices to have Ω1 ∩ Ω3 = ∅ for the existence of minimisers.In a hard penalisation, a sharp interface competitor of area S must be constructed or thehard area constraint should be dropped in favour of a soft term like Sε or (Sε − S)2.• Using [BM10, Theorem 4.1], we could take Bellettini’s approximation of the Helfrich energy

EHelε (u) =

∫Ω

2 + χ

2εv2u,ε −

χ

2ε

∣∣∣∣ε∇2u− W ′(u)

ενu ⊗ νu

∣∣∣∣2 dx

for χ ∈ (−2, 0) in place of the diffuse Willmore energyWε. Here vu,ε is the usual Willmoredensity associated with u and νu = ∇u/|∇u| is the diffuse normal like in Lemma 3.12.• We can use the same modelling techniques for a finite collection of membranes inside

an elastic container. The outer container is modelled by a phase field Uε and the innermembranes by phase fields u1

ε, . . . , uNε . The governing energy is composed by a sum of the

individual elastic energies Eε modified by bending moduli χi > 0, interaction energies Iεpreventing interpenetration and confinement energies Tε:

Fε(Uε, u1ε, . . . u

Nε ) = Eε(Uε) +

N∑i=1

χi Eε(uiε) +1

εβ

N∑i=1

Tε(uiε, Uε)

+1

εβ

N∑i=1

∑j 6=i

Iε(uiε, u

jε)

for some β > 0 where for example

Iε(u, v) =

∫Ω

(u+ 1)2 (v + 1)2 dx

and Tε(u, U) = Iε(u, 2 − U). These energies prevent u, v from being close to +1 at thesame time preventing penetrations between different membranes and u being close to 1where U ≈ −1 enforcing the confinement of u to the elastic container. Surface area can beprescribed for each phase field individually and other modifications can be included. Thedomain Ω can be chosen as a ball large enough to have no influence on the problem. This


appears to be an interesting way to treat the mitochondria-shape problem mentioned inthe introduction.

An interesting open question is whether it is possible to develop to a finer control of the topologyof limit interfaces in three dimensions. While in two dimensions, the only closed connected 1-manifold is the circle, orientable closed surfaces in R3 are fully classified by their genus g ∈ N.This is linked to Gaussian curvature K by the Gauss-Bonnet theorem∫

Σ

K dH2 = 4π (1− g).

Although there are diffuse approximations χε of this invariant [DLW05, DLRW07], investigationsinto their use in prescribing a topology in Willmore problems have not been conducted, neitheranalytically nor numerically.

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Patrick W. Dondl, Department of Mathematical Sciences, Durham University, Durham DH1 1PT,

United Kingdom

E-mail address: [email protected]

Antoine Lemenant, Universite Paris Diderot - Paris 7, U.F.R de Mathematiques, Batiment Sophie

Germain, 75205 Paris Cedex 13, FranceE-mail address: [email protected]

Stephan Wojtowytsch, Department of Mathematical Sciences, Durham University, Durham DH1 1PT,

United KingdomE-mail address: [email protected]

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